A Course in Mathematical Physics 1 and 2

Walter Thirring

A Course in Mathematical Physics

1 and 2

Classical Dynamical Systems and Classical Field Theory

Second Edition

Translated by Evans M. Harrell

With 144 lllustrations

Springer-Verlag New York Wien

Dr. Walter Thirring Institute for Theoretical Physics University of Vienna Austria

Printed on acid-free paper.

© 1992 Springer-Verlag New York Inc.

Dr. Evans M. Harrell The lohns Hopkins University Baltimore, Maryland U.S.A.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA) except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Volume I © 1978, 1992 by Springer-Vedag/Wien Volume 2 © 1979, 1986 by Springer-VedaglWien

Typeset by Asco Trade Typesetting Ltd., Hong Kong.

9 8 7 6 5 4 3 2 1

ISBN-13: 978-0-387-97609-9 DOl: 10.1007/978-1-4684-0517-0

e-ISBN-13: 978-1-4684-0517-0

Summary of Contents

A Course in Mathematical Physics 1: Classical Dynamical Systems, Second Edition

1 Introduction 2 Analysis on Manifolds 3 Hamiltonian Systems 4 Nonrelativistic Motion 5 Relativistic Motion 6 The Structure of Space and Time

A Course in Mathematical Physics 2: Classical Field Theory, Second Edition

1 Introduction 2 The Electromagnetic Field of a Known Charge Distribution 3 The Field in the Presence of Conductors 4 Gravitation

v

Walter Thirring

A Course in Mathematical Physics

1

Classical Dynamical Systems

Translated by Evans M. Harrell

Preface to the Second Edition

The last decade has seen a considerable renaissance in the realm of classical dynamical systems, and many things that may have appeared mathematically overly sophisticated at the time of the first appearance of this textbook have since become the everyday tools of working physicists. This new edition is intended to take this development into account. I have also tried to make the book more readable and to eradicate errors.

Since the first edition already contained plenty of material for a one­semester course, new material was added only when some of the original could be dropped or simplified. Even so, it was necessary to expand the chap­ter with the proof of the K-A-M Theorem to make allowances for the cur­rent trend in physics. This involved not only the use of more refined mathe­matical tools, but also a reevaluation of the word "fundamental." What was earlier dismissed as a grubby calculation is now seen as the consequence of a deep principle. Even Kepler's laws, which determine the radii of the planetary orbits, and which used to be passed over in silence as mystical nonsense, seem to point the way to a truth unattainable by superficial observation: The ratios of the radii of Platonic solids to the radii of inscribed Platonic solids are irrational, but satisfy algebraic equations of lower order. These irrational numbers are precisely the ones that are the least well approximated by rationals, and orbits with radii having these ratios are the most robust against each other's perturbations, since they are the least affected by resonance effects. Some surprising results about chaotic dynamics have been discovered recently, but unfortunately their proofs did not fit within the scope of this book and had to be left out.

In this new edition I have benefited from many valuable suggestions of colleagues who have used the book in their courses. In particular, I am deeply grateful to H. Grosse, H.-R. Griimm, H. Narnhofer, H. Urbantke, and above

IX

x Preface to the Second Edition

all M. Breitenecker. Once again the quality of the production has benefited from drawings by R. Bertlmann and J. Ecker and the outstanding word­processing of F. Wagner. Unfortunately, the references to the literature have remained sporadic, since any reasonably complete list of citations would have overwhelmed the space allotted.

Vienna, July, 1988 Walter Thirring

Preface to the First Edition

This textbook presents mathematical physics in its chronological order. It originated in a four-semester course I offered to both mathematicians and physicists, who were only required to have taken the conventional intro­ductory courses. In order to be able to cover a suitable amount of advanced material for graduate students, it was necessary to make a careful selection of topics. I decided to cover only those subjects in which one can work from the basic laws to derive physically relevant results with full mathematical rigor. Models which are not based on realistic physical laws can at most serve as illustrations of mathematical theorems, and theories whose pre­dictions are only related to the basic principles through some uncontrollable approximation have been omitted. The complete course comprises the following one-semester lecture series:

I. Classical Dynamical Systems II. Classical Field Theory III. Quantum Mechanics of Atoms and Molecules IV. Quantum Mechanics of Large Systems

Unfortunately, some important branches of physics, such as the rela­tivistic quantum theory, have not yet matured from the stage of rules for calculations to mathematically well understood disciplines, and are there­fore not taken up. The above selection does not imply any value judgment, but only attempts to be logically and didactically consistent.

General mathematical knowledge is assumed, at the level of a beginning graduate student or advanced undergraduate majoring in physics or mathe­matics. Some terminology of the relevant mathematical background is

xi

xii Preface to the First Edition

collected in the glossary at the beginning. More specialized tools are intro­duced as they are needed; I have used examples and counterexamples to try to give the motivation for each concept and to show just how far each assertion may be applied. The best and latest mathematical methods to appear on the market have been used whenever possible. In doing this many an old and trusted favorite of the older generation has been forsaken, as I deemed it best not to hand dull and worn-out tools down to the next generation. It might perhaps seem extravagant to use manifolds in a treat­ment of Newtonian mechanics, but since the language of manifolds becomes unavoidable in general relativity, I felt that a course that used them right from the beginning was more unified.

References are cited in the text in square brackets [ ] and collected at the end of the book. A selection of the more recent literature is also to be found there, although it was not possible to compile a complete bibliography.

I am very grateful to M. Breitenecker, J. Dieudonne, H. Grosse, P. Hertel, J. Moser, H. Narnhofer, and H. Urbantke for valuable suggestions. F. Wagner and R. Bertlmann have made the production of this book very much easier by their greatly appreciated aid with the typing, production and artwork.

Walter Thirring

Note about the Translation

In the English translation we have made several additions and corrections to try to eliminate obscurities and misleading statements in the German text. The growing popularity of the mathematical language used here has caused us to update the bibliography. We are indebted to A. Pflug and G. Siegl for a list of misprints in the original edition. The translator is grateful to the Navajo Nation and to the Institute for Theoretical Physics of the University of Vienna for hospitality while he worked on this book.

Evans M. Harrell

Walter Thirring

Contents of Volume 1

Preface to the Second Edition ix

Preface to the First Edition xi

Glossary xv

Symbols Defined in the Text xix

1 Introduction

1.1 Equations of Motion 1.2 The Mathematical Language 4 1.3 The Physical Interpretation 5

2 Analysis on Manifolds 8

2.1 Manifolds 8 2.2 Tangent Spaces 19 2.3 Flows 32 2.4 Tensors 42 2.5 Differentiation 56 2.6' Integrals 61

73

3 Hamiltonian Systems 84

3.1 Canonical Transformations 84 3.2 Hamilton's Equations 91 3.3 Constants of Motion 100 3.4 The Limit t -+ I ± 00 117 3.5 Perturbation Theory: Preliminaries 141 3.6 Perturbation Theory: The Iteration 153

xiii

xiv

4

5

6

Nonrelativistic Motion

4.1 Free Particles 4.2 The Two-Body Problem 4.3 The Problem of Two Centers of Force 4.4 The Restricted Three-Body Problem 4.5 The N-body Problem

Relativistic Motion

5.1 The Hamiltonian Formulation of the Electrodynamic Equations of Motions

5.2 The Constant Field 5.3 The Coulomb Field 5.4 The Betatron 5.5 The Traveling Plane Disturbance 5.6 Relativistic Motion in a Gravitational Field 5.7 Motion in the Schwarzschild Field 5.8 Motion in a Gravitational Plane Wave

The Structure of Space and Time

6.1 The Homogeneous Universe 6.2 The Isotropic Universe 6.3 Me according to Galileo 6.4 Me as Minkowski Space 6.5 Me as a Pseudo-Riemannian Space

Bibliography

Index

Contents

165

165 169 178 186 201

210

210 216 223 229 234 239 245 255

262

262 264 266 268 275

279

283

Glossary

Logical Symbols

V for every :I there exist(s) ~ there does not exist :I! there exists a unique a => b if a then b iff if and only if

Sets

aEA a¢A AuB AnB CA A\B A~B

o C0 AxB

a is an element of A a is not an element of A union of A and B intersection A and B complement of A (In a larger set B: {a: a E B, a¢ A}) {a:aEA,a¢B} symmetric difference of A and B: (A\B) u (B\A) empty set universal set Cartesian product of A and B: the set of all pairs (a, b), a E A, bE B

Important Families of Sets

open sets contains 0 and the universal set and some other specified sets, such that the open sets are closed under union and finite intersection

xv

xvi

closed sets measurable sets

Glossary

the complements of open sets contains 0 and some other specified sets, and

closed under complementation and countable intersection

Borel-measurable sets the smallest family of measurable sets which contains the open sets

null sets, or sets of measure zero the sets whose measure is zero. "Almost everywhere" means" except on a set of measure zero."

An equivalence relation is a covering of a set with a non-intersecting family of subsets. a ~ b means that a and b are in the same subset. An equivalence relation has the prop­erties: i) a ~ a for all a: ii) a ~ b => b ~ a. iii) a ~ b, b ~ c => a ~ c.

Numbers

N 7L IR IR+(IR-) C sup inf I (a, b) [a,b] (a,b] and [a, b) IR"

natural numbers integers real numbers positive (negative) numbers complex numbers supremum, or lowest upper bound infimum, or greatest lower bound any open interval the open interval from a to b the closed interval from a to b half-open intervals from a to b IRx···xlR '-.--'

N times This is a vector space with the scalar product (Yb ... , YNlx l , ••• , XN) = Li: I YiXi

Maps ( = Mappings, Functions)

f:A -+ B f(A) rl(b)

r l

rl(B) f is injective (one-to-one) f is surjective (onto) f is bijective

1

for every a E A an element f(a) E B is specified image of A, i.e., if f:A -+ B, {f(a) E B:a E A} inverse image of b, i.e. {a E A: f(a) = b} inverse mapping to f. Warning: 1) it is not necessarily a

function 2) distinguish from 11f when B = IR inverse image of B: UbEB f-I(b) al oF a2 => f(ad oF f(a2) f(A) = B f is injective and surjective. Only in this case is f - I a

true function the function defined from A I X A 2 to B I X B 2, so that

(a\> a2) -+ (fl (al),fz(a2» fl composed with f2: if fl : A -+ Band f2: B -+ C, then f2 0 fl:A -+ C so that a -+ f2(fI(a»

identity map, when A = B; i.e., a -+ a. Warning: do not confuse with a --+ 1 when A = B = R

Glossary

fl. fl. f is continuous f is measurable suppf

C' Co XA

Topological Concepts

Topology compact set

connected set

discrete topology trivial topology

simply connected set

(open) neighborhood of a E A

f restricted to a subset V c A e~aluation of the map f at the point a; i.e.,f(a). the inverse image of any open set is open

xvii

the inverse image of any measurable set is measurable support off: the smallest closed set on whose

complement f = 0 the set of r times continuously differentiable functions the set of C' functions of compact (see below) support characteristic function of

A:xia) = 1. ..

any family of open sets, as defined above a set for which any covering with open sets has

a finite subcovering a set for which there are no proper subsets which

are both open and closed the topology for which every set is an open set the topology for which the only open sets are

o andC0 a set in which every path can be continuously

deformed to a point any open subset of A containing a. Usually

denoted by V or V. (open) neighborhood ofB c A p is a point of accumulation

any open subset of A containing B for any neighborhood V containing p,

(= cluster point) B B is dense in A B is nowhere dense in A metric (distance function) for A

separable space homeomorphism product topology on Al x A2

Mathematical Conventions

Of/Oqi

dq(r)

dt

V nB"# {p} or 0 closure of B: the smallest closed set containing B B=A A\B is dense in A a map d: A x A -+ IR such that dCa, a) = 0;

dCa, b) = deb, a) > 0 for b "# a; and dCa, c) :;; dCa, b) + deb, c) for all a, b, c in A. A metric induces a topology on A, in which all sets of the form {b:d(b, a) < II} are open.

a space with a countable dense subset a continuous bijection with a continuous inverse the family of open sets of the form V I X V 2,

where VI is open in Al and V 2 is open in A2 ,

and unions of such sets

q(t)

detlMijl TrM

determinant of the matrix Mij

c5f, c5ij LiMii

I if i = j, otherwise 0

xviii Glossary

the totally antisymmetric tensor of degree m. with values ± 1.

Mt

M* transposed matrix: (Mt)ij = Mji

V· w, (vlw), or (v· w)

v x w or [v 1\ w] Vf

Hermitian conjugate matrix: (M*)ij = (Mji)* scalar (inner, dot) product cross product gradient of f

V x f V ·f

curl off divergence of f

IIvil (in 3 dimensions, Ivl) ds

length of the vector v: Ilv II = (Ii = 1 vl)1/2 = d(O. v) differential line element

dS dmq .1 II 4..

dQ

Mat.(IR) O(x)

differential surface element m-dimensional volume element is perpendicular (orthogonal) to is parallel to angle element of solid angle

the set of real n x n matrices order of x

The summation convention for repeated indices is understood except where it does not make sense. For example. Li1Xl stands for Ik Li1Xl.

Groups

GL. group of n x n matrices with nonzero determinant O. group of n x n matrices M with M Mt = 1 (unit matrix) SO. subgroup of O. with determinant 1 E. Euclidean group S. group of permutations of n elements U. group of complex n x n matrices M with M M* = 1 (unit matrix)

SP. group of symplectic n x n matnces

Physical Symbols

mi Xi

t = xO/c S

qi Pi ei K

c Ii = h/21t F; g./I E B

mass of the i-th particle Cartesian coordinates of the i-th particle time proper time generalized coordinates generalized momenta charge of the i-th particle gravitational constant speed of light Planck's constant divided by 21t electromagnetic field tensor gravitational metric tensor (relativistic gravitational potential) electric field strength magnetic field strength in a vacuum is on the order of is much greater than

Symbols Defined in the Text

Df derivative of f: JR" -+ JR"' (2.1.1 ) (V. <II) chart (2.1.3) T" n-dimensional torus (2.1.7; 2) S" n-dimensional sphere (2.1.7; 2) oM boundary of M (2.1.20) 0C<q) mapping of the tangent space into JR"' (2.2.1) 1'q(M) tangent space at the point q (2.2.4) Tq(f) derivative o{ f at the point q (2.2.7) T(M) tangent bundle (2.2.12) n projection onto a basis (2.2.15) T(f) derivative of f: M I -+ M 2 (2.2.17) .rMM) set of vector fields (2.2.19) <II. induced mapping on .r~ (2.2.21) Lx Lie derivative (2.2.25; I), (2.5.7) OJ natural basis on the tangent space (2.2.26) <II'" I flow (2.3.7) r'" I automorphism of a flow (2.3.8) W action (2.3.16) L Lagrangian (2.3.17) H Hamiltonian (2.3.26) T:(M) cotangent space (2.4.1) e'!' , dual basis (2.4.2; I) df differential of a function (2.4.3; I) Tq~(M) space of tensors (2.4.4) ® tensor product (2.4.5) T~(M) tensor bundle (2.4.8) .r~(M) set of tensor fields (2.4.28) tif differential of a function (2.4.13; 1) g pseudo-Riemannian metric (2.4.27) • fiber product (2.4.34) x

xix

xx Symbols Defined in the Text

T*(<1» transposed derivative (2.4.34) (<1>- 1)* pull-back, or inverse image of the covariant tensors (2.4.41) Ep(M) set of p-forms (2.4.27) A wedge (outer, exterior) product (2.4.28) * *-mapping (2.4.18)

ix interior product (2.4.33) d exterior derivative (2.5.1 ) [ ] Lie bracket (2.5.9; 6) 0,w canonical forms (3.1.1 ) Q Liouville measure (3.1.2;3) XH Hamiltonian vector field (3.1.9) b bijection associated with w (3.1.9) { } Poisson brackets (3.1.11)

Me generalized configuration space (3.2.12) Yf Hamiltonian on Me (3.2.12) (I, cp) action-angle variables (3.3.14)

Q± M011er transformations (3.4.4) S scattering matrix (3.4.9) da differential scattering cross-section (3.4.12) L angular momentum (4.1.3) K boost (4.1.9)

'l,p Minkowski space metric (5.1.2)

" I/JI - v21c2 (relativistic dilatation) (5.1.2)

F electromagnetic 2-form (5.1.10; I) A I-form of the potential (5.1.10; I) /\ Lorentz transformation (5.1.12)

ro Schwarzschild radius (5.7.1)

Walter Thirring

A Course in Mathematical Physics

2

Classical Field Theory

Second Edition

Translated by Evans M. Harrell

Preface

In the past decade the language and methods of modern differential geometry have been increasingly used in theoretical physics. What seemed extravagant when this book first appeared 12 years ago, as lecture notes, is now a commonplace. This fact has strengthened my belief that today students of theoretical physics have to learn that language-and the sooner the better. After all, they will be the professors of the twenty-first century and it would be absurd if they were to teach then the mathematics of the nineteenth century. Thus for this new edition I did not change the mathematical language. Apart from correcting some mistakes I have only added a section on gauge theories. In the last decade it has become evident that these theories describe fundamental interactions, and on the classical level their structure is suffi­ciently clear to qualify them for the minimum amount of knowledge required by a theoretician. It is with much regret that I had to refrain from in­corporating the interesting developments in Kaluza-Klein theories and in cosmology, but I felt bound to my promise not to burden the students with theoretical speculations for which there is no experimental evidence.

I am indebted to many people for suggestions concerning this volume. In particular, P. Aichelburg, H. Rumpf and H. Urbantke have contributed generously to corrections and improvements. Finally, I would like to thank Dr. I. Dahl-lensen for redoing some of the figures on the computer.

Vienna December, 1985

W. Thirring

iii

Contents of Volume 2

Symbols Defined in the Text vii

1 Introduction

2

3

4

1.1 Physical Aspects of Field Dynamics 1.2 The Mathematical Formalism 1.3 Maxwell's and Einstein's Equations

The Electromagnetic Field of a Known Charge Distribution

I 10 28

46

2.1 The Stationary-Action Principle and Conservation Theorems 46 2.2 The General Solution 57 2.3 The Field of a Point Charge 71 2.4 Radiative Reaction 89

The Field in the Presence of Conductors 102

3.1 The Superconductor 102 3.2 The Half-Space, the Wave-Guide, and the Resonant Cavity 113 3.3 Diffraction at a Wedge 125 3.4 Diffraction at a Cylinder 138

Gravitation

4.1 Covariant Differentiation and the Curvature of Space 4.2 Gauge Theories and Gravitation

155

155 174

v

VI

4.3 Maximally Symmetric Spaces 4.4 Spaces with Maximally Symmetric Submanifolds 4.5 The Life and Death of Stars 4.6 The Existence of Singularities

Bibliography

Index

Contents

191 204 223 236

253

259

Symbols Defined in the Text

ei1i2 , .. ip basis of the p-forms ( 1.2.3) Ep(M) linear space of the p-forms (1.2.5; 2)

d exterior differential (1.2.6) (1)[,'1 restriction of a form (1.2.7; 3)

E~(U) space of m-forms with compact support ( 1.2.9)

<ei(x)Id'(x» scalar product ( 1.2.14)

iv interior product (1.2.16)

* isomorphism between Ep and Em - p ( 1.2.17)

6 codifferential (1.2.19) .1 Laplace-Beltrami operator ( 1.2.20)

Lv Lie derivative ( 1.2.23)

w\ affine connection ( 1.2.25)

(Uik affine connection ( 1.2.25) 0(x) Heaviside step function ( 1.2.31) 6(x) Dirac delta function (1.2.31)

6 .. Dirac delta form ( 1.2.33)

Gx Green function ( 1.2.35)

E,B,F electric and magnetic fields (1.3. J) A vector potential (1.3. 7)

1\ gauge function (1.3.10; I)

J current (1.3.12)

Q total charge (1.3.18; 2) T'P energy-momentum tensor ( 1.3.20) p' total energy-momentum (1.3.21 )

.OJ' energy-momentum form of the field ( 1.3.22) z(s) world-line (1.3.25; 2) t' energy-momentum form of matter (1.3.25; 2) !f Lagrangian (2.1.1 )

W action (2.1.1)

vii

viii Symbols Defined in the Text

9' Poynting's vector (2.1.13) D±(N) domains of influence (2.1.15)

Di/ Green function (2.2.5) Drel(x) retarded Green function (2.2.7) G~el

x retarded Green function (form) (2.2.7) Frel retarded field strength (2.2.9) Fin incoming field strength (2.2.15) Foul outgoing field strength (2.2.15) Frad radiation field (2.2.21) D(x) D-function (2.2.22) (jE energy loss per period (2.4.4; 2)

j specified current (3.1.7) e(k) dielectric constant (3.1.19; 3) S superpotential (3.1.21; 1)

F(z) Fresnel's integral (3.3.10)

<I) scalar product (4.1.3) Sp Sections (4.1.9) D exterior covariant derivative (4.1.10) Dx covariant derivative (4.1.15) n curvature form (4.1.19) R curvature in space-time (4.1.20; 1)

f ijk Christoffel symbol (4.1.36) Rijkm Riemann-Christoffel tensor (4.1.44; 2) C jk Weyl forms (4.1.44; 3)

ro Schwarzschild radius (4.4.41) K curvature parameter (4.4.42) c rate of convergence (4.6.8) rex) future of x (4.6.18(a» r(x) past of x (4.6. I 8(a» C(x, S) set of causal curves (4.6.18(b» C1(x; S) set of differentiable causal curves (4.6. 18(b» d(.l.) length of .l. (4.6.l8(c»